Solution Found!
Let p(y) denote the probability function associated with a
Chapter 3, Problem 142E(choose chapter or problem)
Let \(p(y)\) denote the probability function associated with a Poisson random variable with mean \(\lambda\)
a. Show that the ratio of successive probabilities satisfies \(\frac{p(y)}{p(y-1)}=\frac{\lambda}{y}\), for \(y=1\), 2,....
b. For which values of is \(p(y)>p(y-1)\)?
c. Notice that the result in part (a) implies that Poisson probabilities increase for awhile as
increases and decrease thereafter. Show that \(p(y)\) maximized when = the greatest integer
less than or equal to \(\lambda\).
Equation Transcription:
Text Transcription:
p(y)
lambda
p(y) over p(y-1)=lambda over y
y=1
p(y)>p(y-1)
p(y)
lambda
Questions & Answers
QUESTION:
Let \(p(y)\) denote the probability function associated with a Poisson random variable with mean \(\lambda\)
a. Show that the ratio of successive probabilities satisfies \(\frac{p(y)}{p(y-1)}=\frac{\lambda}{y}\), for \(y=1\), 2,....
b. For which values of is \(p(y)>p(y-1)\)?
c. Notice that the result in part (a) implies that Poisson probabilities increase for awhile as
increases and decrease thereafter. Show that \(p(y)\) maximized when = the greatest integer
less than or equal to \(\lambda\).
Equation Transcription:
Text Transcription:
p(y)
lambda
p(y) over p(y-1)=lambda over y
y=1
p(y)>p(y-1)
p(y)
lambda
ANSWER:
Solution
Step 1 of 3
a) We have to show that
The pmf of poisson distribution is
And
Now
=
=
Hence we prove that